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An optimal property of Chebyshev expansions
JOURNAL
OF APPROXIMATION
An Optimal
THEORY
2,312-317 (1969)
Property
of Chebyshev
Expansions
T. J. RIVLIN AND M. WAYNE WILSON IBM, Thomas J. Watson Research Center, Yorktown Heights, New York 10598 INTRODUCTION
Chebyshev polynomials are extremely popular in numerical analysis. One of their virtues is that expansions of functions in series of Chebyshev polynomials are thought to converge more rapidly than expansions in series of other orthogonal polynomials, and some supporting asymptotic evidence for this belief is presented in Lanczos [2]. Our purpose here is to demonstrate that for a certain restricted class of functions, the truncated Chebyshev expansion is best in some fairly large class of Jacobi expansions, and, thus, to provide further solid foundation for the Chebyshevfaith. The remainder of the Introduction is devoted to presenting notation and setting the stage. In Section 1, we make precise the sensein which Chebyshev expansions are best, while Section 2 is given over to various counter-examples to the results in Section 1. Let Pp, s’(x) be the Jacobi polynomials with CL, ,l3> -1 (that is, the orthogonal polynomials on I: [-1, l] with respect to the weight function w(oz,/~;x)= (1 - x)a(l +x)“), normalized in the usual fashion (cf. Szegii [4], p. 58)). For each (cc,/I), (y, 8) we have Pp “‘(x) = i bjk(u”,p; y, s)Py “‘(x), j=o and we adopt the usagethat bj, = 0 if j > k. We also adopt the convention that pp “‘(x) = Xk
(1)
and admit the values cc= ,6= ~0,with this definition in mind. If (a,/?), (7,s) are such that j=O,..., k;k=0,1,2 ,..., bj.da, fl; Y, S>> 0, (2) we say that Condition P holds. It is known that if (i) /3=8andcr>y, or (ii) M.= /3,y = 6 and a > y, then Condition P holds. (See,for example, Askey [I] and Rainville [3]. Askey also gives some other conditions on the indices which imply Condition P.) 312
CHEBYSHEVEXPANSIONS
31
To eachf E C(I) we associateits “Fourier” coefficients I l fda,P) = hj(E, p> s -l f(x) Pp B’(x) w(a, p; x) dx, where hj(a,p)=yl andj=0,1,2,....
[P~“~~)(x)~‘w(a,~;x)dx
(Of course, here we assumea < co.) We put s?yyx) =f+:: fj(% p>pp Yx3, up “‘(x) =f(x) - s,p “‘(x)
and &A@& PI = IIrP q 9 where jj aI/is the uniform norm on I and k = 0, I.,&. e.. We sayfE U(cz,/$ if9 and only if, f(x) = J.pxa, P>W%),
w
uniformly in I, whilef E A(a, ,kI)if, and only if, the seriesis absolutely convergent for each x in 1. In caseM.= /I = co,(3’) is assumedto be the Taylor expansive about the origin, and sofj(cc,/3)are the Taylor coefhcients. 1. MAIN RESULTS
Our results are based on the following simple LEMMA.~~~E U(ct,p) andyc
co then
f(x) P JV’ “‘(x) w(y, 6 ; x) dx =&)
j;, J
3
P!“, B)(x)PQ’* J @(x) w(y, 6; x) dx
(The term-by-term integration is justified by the uniform convergence.)
314
RIVLINAND
WILSON
An immediate consequenceof the lemma is THEOREM
1. Iff E U(a,j3),fk(~~,fi)> 0, k = 0, 1,2,. . ., and Condition P holds,
then fj(y,S)>O,
j=o,1,2 )....
The lemma also leads to the following convergenceresults, which, although not neededin what follows, is stated here for its own interest. THEOREM
2. Suppose (4)
6 =zy with y 2 -3 and Condition P holds. Then f E WY, 8) fl 4% 3. Proof. Since y > 6 and y 2 -+, we know (&ego [4], p. 166) that $Iy.;
j=o,1,2
)....
(5)
k = 0, 1,2, . . ., -I”ca;tyclIPp,“‘(x)l = Pp,s)(l), (6) . . and hence, in view of (4) and the Weierstrass M-Testfe U(a,/3). As a further consequenceof (5), the Theorem will be proved if we can show that the sequence
Fm= j$oIh(r, @IJ’jya6)W is bounded. SincefE U(tc,p), the lemma together with Condition P give
j=o
k=j
Thus
= kzoIAc(%P)I PP W) < a> by (4).
CHEBYSHEVEXPANSIONS
315
We turn now to our main result. THEOREM 3. Suppose
.fxa, P) a 0,
k>n,
m
6 < y with y 2 -3, Condition P holds and
ProoJ As we saw in the proof of Theorem 2, the hypotheses of our Theorem imply that (6) holds and hence that
Ma, P>= ,=~+,X(a, tWP p8’tl> = ,=z+,.fXa, t3
the exchange of summations being justified since all summands are positive (cf. Titchmarsh [5], Ch. I). Hence,
in view of Theorem 1 and the Lemma ((8) implies that f E U(a,j3)>.Finally, (7), Condition P and the fact that Pjr, @(1) > 0,j = 0, ‘. ., n, conclude the proof. Remark 1. If in Theorem 3, in place of Condition P we assumethat (i) p = 8 and a >y, or (ii) a = p > y = 6, then we can conclude that equality holds in (9) if, and only if,fis a polynomial of degreein. For, iff,(a,/3) > 0 for some D-Z > n, then either b&a,/?; y, 6) > 0 or bl,(a,P; y, 8) > 0 and so
,j+l .&(a, PI b.da, 13;Y, 8) > 0 for eitherj= 0 orj=
1.
316
RIVLIN AND WILSON
Remark 2. Consider the ultraspherical case(a = fl; y = 8) and assume (-Qkh(% 4 2 0;
k>n
in place of (7). Then since Theorem 3 can now be applied tof(-x), remains true forf(x).
(10) Theorem 3
2. SOME EXAMPLES Theorem 3 is, perhaps, most interesting in the ultraspherical case (a = 8; y = 8) since th is f amily of polynomials includes those bearing the names of Legendre and Chebyshev (both kinds), as well as, in our presentation, Taylor. In the ultraspherical case, Theorem 3 is valid if 00 y > --+. (Recall that y = --$ corresponds to the Chebyshev expansion.) We shall next present several examples which show that we cannot dispense with requirements (7) or (IO), nor demonstrate that the expansion in Chebyshev polynomials produces the smallest error in the somewhat larger class, a > y ~-1. We suppress the second index in what follows, since we shall deal only with the ultraspherical case. Examples 1. Takef(x) = x3 - $x2 - x and suppose a = co,n = 0. Neither (7) nor (10) holds and tin ROW = ROWI --*
where b2-WI Y=4(1 + Ibj)-,,2-” 3. f(x)
I<
-3.
= x3 - x2, CC = co,IZ= 1. Now (10) holds, yet -,yynGmMY) = RI(~)
where -l
(7 - -.543). REFERENCES
1. R. ASKEY, Jacobi polynomial expansions with positive coefficients and imbeddings of projective spaces. Bull. Amer. Math. Sot. 74 (1968), 301-304.
CHEBYSHEV EXPANSIONS
2. C. LANCZOS, “Introduction to Tables of Chebyshev Polynomials S,(x) and C,(x),” Applied Math. Series 9, National Bureau of Standards, U.S. Government Printing Office, Washington, D.C., 1952. 3. E. D. RAINVILLE, “Special Functions.” Macmillan, N.Y., 1960. 4. C. SZEG~, “Orthogonal Polynomials.” AMS, Colloq. Publ., vol. XXIII, AMS, N.Y., 1959. 5. E. C. TITCHMARSH,“The Theory of Functions,” 2nd ed. Oxford University Press, 6950.
OF APPROXIMATION
An Optimal
THEORY
2,312-317 (1969)
Property
of Chebyshev
Expansions
T. J. RIVLIN AND M. WAYNE WILSON IBM, Thomas J. Watson Research Center, Yorktown Heights, New York 10598 INTRODUCTION
Chebyshev polynomials are extremely popular in numerical analysis. One of their virtues is that expansions of functions in series of Chebyshev polynomials are thought to converge more rapidly than expansions in series of other orthogonal polynomials, and some supporting asymptotic evidence for this belief is presented in Lanczos [2]. Our purpose here is to demonstrate that for a certain restricted class of functions, the truncated Chebyshev expansion is best in some fairly large class of Jacobi expansions, and, thus, to provide further solid foundation for the Chebyshevfaith. The remainder of the Introduction is devoted to presenting notation and setting the stage. In Section 1, we make precise the sensein which Chebyshev expansions are best, while Section 2 is given over to various counter-examples to the results in Section 1. Let Pp, s’(x) be the Jacobi polynomials with CL, ,l3> -1 (that is, the orthogonal polynomials on I: [-1, l] with respect to the weight function w(oz,/~;x)= (1 - x)a(l +x)“), normalized in the usual fashion (cf. Szegii [4], p. 58)). For each (cc,/I), (y, 8) we have Pp “‘(x) = i bjk(u”,p; y, s)Py “‘(x), j=o and we adopt the usagethat bj, = 0 if j > k. We also adopt the convention that pp “‘(x) = Xk
(1)
and admit the values cc= ,6= ~0,with this definition in mind. If (a,/?), (7,s) are such that j=O,..., k;k=0,1,2 ,..., bj.da, fl; Y, S>> 0, (2) we say that Condition P holds. It is known that if (i) /3=8andcr>y, or (ii) M.= /3,y = 6 and a > y, then Condition P holds. (See,for example, Askey [I] and Rainville [3]. Askey also gives some other conditions on the indices which imply Condition P.) 312
CHEBYSHEVEXPANSIONS
31
To eachf E C(I) we associateits “Fourier” coefficients I l fda,P) = hj(E, p> s -l f(x) Pp B’(x) w(a, p; x) dx, where hj(a,p)=yl andj=0,1,2,....
[P~“~~)(x)~‘w(a,~;x)dx
(Of course, here we assumea < co.) We put s?yyx) =f+:: fj(% p>pp Yx3, up “‘(x) =f(x) - s,p “‘(x)
and &A@& PI = IIrP q 9 where jj aI/is the uniform norm on I and k = 0, I.,&. e.. We sayfE U(cz,/$ if9 and only if, f(x) = J.pxa, P>W%),
w
uniformly in I, whilef E A(a, ,kI)if, and only if, the seriesis absolutely convergent for each x in 1. In caseM.= /I = co,(3’) is assumedto be the Taylor expansive about the origin, and sofj(cc,/3)are the Taylor coefhcients. 1. MAIN RESULTS
Our results are based on the following simple LEMMA.~~~E U(ct,p) andyc
co then
f(x) P JV’ “‘(x) w(y, 6 ; x) dx =&)
j;, J
3
P!“, B)(x)PQ’* J @(x) w(y, 6; x) dx
(The term-by-term integration is justified by the uniform convergence.)
314
RIVLINAND
WILSON
An immediate consequenceof the lemma is THEOREM
1. Iff E U(a,j3),fk(~~,fi)> 0, k = 0, 1,2,. . ., and Condition P holds,
then fj(y,S)>O,
j=o,1,2 )....
The lemma also leads to the following convergenceresults, which, although not neededin what follows, is stated here for its own interest. THEOREM
2. Suppose (4)
6 =zy with y 2 -3 and Condition P holds. Then f E WY, 8) fl 4% 3. Proof. Since y > 6 and y 2 -+, we know (&ego [4], p. 166) that $Iy.;
j=o,1,2
)....
(5)
k = 0, 1,2, . . ., -I”ca;tyclIPp,“‘(x)l = Pp,s)(l), (6) . . and hence, in view of (4) and the Weierstrass M-Testfe U(a,/3). As a further consequenceof (5), the Theorem will be proved if we can show that the sequence
Fm= j$oIh(r, @IJ’jya6)W is bounded. SincefE U(tc,p), the lemma together with Condition P give
j=o
k=j
Thus
= kzoIAc(%P)I PP W) < a> by (4).
CHEBYSHEVEXPANSIONS
315
We turn now to our main result. THEOREM 3. Suppose
.fxa, P) a 0,
k>n,
m
6 < y with y 2 -3, Condition P holds and
ProoJ As we saw in the proof of Theorem 2, the hypotheses of our Theorem imply that (6) holds and hence that
Ma, P>= ,=~+,X(a, tWP p8’tl> = ,=z+,.fXa, t3
the exchange of summations being justified since all summands are positive (cf. Titchmarsh [5], Ch. I). Hence,
in view of Theorem 1 and the Lemma ((8) implies that f E U(a,j3)>.Finally, (7), Condition P and the fact that Pjr, @(1) > 0,j = 0, ‘. ., n, conclude the proof. Remark 1. If in Theorem 3, in place of Condition P we assumethat (i) p = 8 and a >y, or (ii) a = p > y = 6, then we can conclude that equality holds in (9) if, and only if,fis a polynomial of degreein. For, iff,(a,/3) > 0 for some D-Z > n, then either b&a,/?; y, 6) > 0 or bl,(a,P; y, 8) > 0 and so
,j+l .&(a, PI b.da, 13;Y, 8) > 0 for eitherj= 0 orj=
1.
316
RIVLIN AND WILSON
Remark 2. Consider the ultraspherical case(a = fl; y = 8) and assume (-Qkh(% 4 2 0;
k>n
in place of (7). Then since Theorem 3 can now be applied tof(-x), remains true forf(x).
(10) Theorem 3
2. SOME EXAMPLES Theorem 3 is, perhaps, most interesting in the ultraspherical case (a = 8; y = 8) since th is f amily of polynomials includes those bearing the names of Legendre and Chebyshev (both kinds), as well as, in our presentation, Taylor. In the ultraspherical case, Theorem 3 is valid if 00 y > --+. (Recall that y = --$ corresponds to the Chebyshev expansion.) We shall next present several examples which show that we cannot dispense with requirements (7) or (IO), nor demonstrate that the expansion in Chebyshev polynomials produces the smallest error in the somewhat larger class, a > y ~-1. We suppress the second index in what follows, since we shall deal only with the ultraspherical case. Examples 1. Takef(x) = x3 - $x2 - x and suppose a = co,n = 0. Neither (7) nor (10) holds and tin ROW = ROWI --*
where b2-WI Y=4(1 + Ibj)-,,2-” 3. f(x)
I<
-3.
= x3 - x2, CC = co,IZ= 1. Now (10) holds, yet -,yynGmMY) = RI(~)
where -l
(7 - -.543). REFERENCES
1. R. ASKEY, Jacobi polynomial expansions with positive coefficients and imbeddings of projective spaces. Bull. Amer. Math. Sot. 74 (1968), 301-304.
CHEBYSHEV EXPANSIONS
2. C. LANCZOS, “Introduction to Tables of Chebyshev Polynomials S,(x) and C,(x),” Applied Math. Series 9, National Bureau of Standards, U.S. Government Printing Office, Washington, D.C., 1952. 3. E. D. RAINVILLE, “Special Functions.” Macmillan, N.Y., 1960. 4. C. SZEG~, “Orthogonal Polynomials.” AMS, Colloq. Publ., vol. XXIII, AMS, N.Y., 1959. 5. E. C. TITCHMARSH,“The Theory of Functions,” 2nd ed. Oxford University Press, 6950.